THE COHOMOLOGY FLAT PRODUCT AND THE BERSTEIN ALGEBRA OF A CO-H-SPACE

Abstract. Let X be a 1{connected cogroup-like space. If R is a ring, then a coho-mology (flat) product Hp+1(X;R) ⊗Hq+1(X;R) ! Hp+q+1(X;R) was dened in [Ar1]. If we set Ap(X;R) = Hp+1(X;R) for p> 0 and A0(X;R) = R, then A(X;R) is a graded algebra. In [Be] a coalgebra B(X;K) and dual algebra B(X;K) were de ned for X a cogroup-like space and K a eld. Our main result is that A(X;K) and B(X;K) are isomorphic algebras for X of nite type over K. It follows that the conilpotency class of X is bounded below by the length of the longest product in the algebras B(X;K). In 1964 Arkowitz introduced a product of homotopy sets which assigns an el-ement f; g 2 [X;A [ B] to elements 2 [X;A] and 2 [X;B], where X is a 1{connected associative co-H-space and A [ B is the bre of A_B! AB [Ar1]. Since X is a cogroup-like space [Ar2, Cor. 1.11], we can form the commutator 2 of the inclusions i1: X! X _X as rst summand and i2: X! X _X as second summand in the group of homotopy classes [X;X _X]. We let ̂2: X! X [ X be the unique lifting of 2 into the bre and set f; g = ( [ ) ̂2. The produc